All Questions
Tagged with cosmology metric-tensor
180
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Is the tensor product involved in the metric a symmetric product?
The expression of the FRW metric in Cosmology in usually written as:
$$ds^2=-dt^2+a^2(t)d\vec{x}^2$$
where $c=1$. However, $dt^2$ is a shortening of $dt\otimes dt$, that is, of the tensor product of $...
- 293
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36
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Question coming from Cosmological Perturbation
We consider the following scalar perturbation on the FRW metric:
$$ ds^2 = -(1 + 2\phi)dt^2 +2a\partial_i B dx^i dt + a^2 \left( (1 - 2\psi)\delta_{ij} + 2\partial_{ij}E\right) dx^i dx^j $$
where $\...
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Is there a metric, a solution to Einstein's field equations, for a single body in a space of uniform non-zero density?
The Swarzschild metric describes a single body in an empty space with zero density, while the FLRW metric is presumably for a space with uniform non-zero density but no single body. But is there a ...
- 131
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Is the FRW metric, based on spatial homogeneity and isotropy, rotationally and translationally invariant? If so, how?
The spatial part of the Minkowski metric, written in the Cartesian coordinates, $$d\vec{ x}^2=dx^2+dy^2+dz^2,$$ is invariant under spatial translations: $\vec{x}\to \vec{x}+\vec{a}$, where $\vec{a}$ ...
- 11.8k
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110
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Saddle Shaped Universe
The universe, as described by FLRW metric, if $k = -1$ is clearly a 2 sheet 3-hyperboloid described by $x^2+y^2+z^2-w^2=-R^2$. So where does the more common saddle shaped picture of the open universe ...
- 1,161
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68
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Homogeneous and Isotropic But not Maximally Symmetric Space
Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
- 1,161
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3
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199
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Change of variables from FRW metric to Newtonian gauge
My question arises from a physics paper, where they state that if we take the FRW metric as follows, where $t_c$ and $\vec{x}$ are the FRW comoving coordinates:
$$ds^2=-dt_c^2+a^2(t_c)d\vec{x}_c^2$$
...
- 293
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27
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A question about Friedmann cosmological expansion equation
A recent paper, arXiv:2403.01555, gives the equations for homogeneity and isotropy of an expanding 3-space as expressed in the following
distance interval as $x^i = (t, \chi, \theta, \phi)$ and $x^i + ...
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1
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66
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Name of metric used by Friedmann
In his original paper, Friedmann used the following dynamic and symmetrical metric:
$$\mathrm{d}s^2=a(t)^2\left(\mathrm{d}\chi^2+\sin (\chi)^2\left(\mathrm{d}\theta^2+\sin (\theta)^2 \mathrm{d}\phi^2\...
- 181
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120
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Transformation under coordinate transformation of scalar perurbation of FLRW metric
For the past few days I've been studying perturbation in cosmology. More specifically I am now busy with chapter 6 in Dodelson's Modern cosmology. In this book the perturbed FLRW metric which only ...
- 29
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60
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Robertson-Walker metric exercise [closed]
I'm trying to solve an exercise from my astrophysics and cosmology class, the request is the following, starting from the RW metric expression:
$$ \begin{equation*}
ds^2=c^2 dt^2 - a^2 \left ( \frac{...
- 41
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88
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Why is $h_{\mu\nu}$ not a tensor in the perturbed Universe in cosmological perturbation theory?
In the cosmological perturbation theory course per Hannu Kurki-Suonio (2022) : https://www.mv.helsinki.fi/home/hkurkisu/CosPer.pdf, there is a remark in the text page 5 that puzzles me. The text goes ...
- 1,109
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1
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148
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Deriving Klein-Gordon equation in curved spacetime [closed]
I try to drive The Klein-Gordon equation for a massless scalar field in case of FRW metric:
$$
ds^2= a^2(t) [-dt^2 + dx^2]
$$
So I start by:
$$\left(\frac{1}{g^{1/2}}\partial_{\mu}(g^{1/2}g^{\mu\nu}\...
- 395
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170
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Why isn't the curvature scale in Robertson-Walker metric dynamic?
$$ds^2=-c^2dt^2+a(t)^2 \left[ {dr^2\over1-k{r^2\over R_0^2}}+r^2d\Omega^2 \right]$$
This is the FRW metric, here k=0 for flat space, k=1 for spherical space, k=-1 for hyperbolic space. $R_0$ is the ...
- 619
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230
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Deriving the Ricci tensor on the flat FLRW metric
I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric
$$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find:
$$R_{ij} = \delta_{ij}[2\...
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